Functions:Injective

A function $f:A\to B$ is injective, or "one-to-one", if for all $a_{1},a_{2}\in A$, $a_{1}\neq a_{2}$ implies that $f(a_{1})\neq f(a_{2})$ (or for all $a_{1},a_{2}\in A$, $f(a_{1})=f(a_{2})$ implies that $a_{1}=a_{2}$). That is, a function is injective if each output is unique to a specific input, and no two distinct inputs map to the same output.

Examples:

• Let $A=\{1,2,3\}$ and $B=\{1,2,3\}$, and let $f:A\to B$ such that $f(1)=2$, $f(2)=1$, and $f(3)=3$. $f$ is an injective function because each output of $f$ is mapped to by exactly one input.

• Let $g:A\to B$ such that $g(1)=1$, $g(2)=1$, and $g(3)=2$. $g$ is not an injective function since $g(1)=g(2)$.

• $f:\mathbb {R} \to \mathbb {R}$, $f(x)=x$ is an injective function, since $x_{1}\neq x_{2}\implies f(x_{1})\neq f(x_{2})$ for all $x_{1},x_{2}\in \mathbb {R}$.

• Let $f:\mathbb {R} \to \mathbb {R}$, $f(x)=x^{2}$. This function is NOT injective because for $x\neq 0$, $f(x)=x^{2}=(-x)^{2}=f(-x)$, but $x\neq -x$. For example, $1\neq -1$ while $f(1)=f(-1)=1$, which conflicts with the definition of injectivity.

Resources

• Injective Function, Wikipedia
• Function Types, OpenStax
• Proofs and Fundamentals: A First Course in Abstract Mathematics, pages 154-164

Also see functions.